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\pagestyle{myheadings} \markright{Math 4310 (Fall 2020) \hspace{2in}
(HW6)}

\begin{document}


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  {\large Homework \# 8

    \medskip
    Math 4310 Fall 2020

    \medskip
    
    Due Friday, 11/6/20
  }

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Please submit your completed homework via gradescope on canvas.
I encourage you to work with your classmates on this homework (except for the journal entries!)
When you submit your work, please {\bf list your collaborators}.
(Your grade will not be affected.)
Even if you work in a group, you should write up 
your solutions {\bf yourself}! 
You should include all computational details, and proofs should be carefully 
written with full details.
As always, please write {\bf neatly and legibly} (feel free to use \LaTeX\ to write up your solutions, if you wish!).

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\noindent {\bf Journal entry.} There is no journal entry this week.

\vskip 0.1in

\noindent {\bf Exercises.} 

\vskip 0.1in

\begin{enumerate}

\item For each of the following matrices or linear maps (over the given field),
  find the eigenvalues and eigenvectors. Decide if the matrix or linear map is
  diagonalizable.
  \babc
\item $A = \begin{pmatrix} 1 & 1 \\ 0  & 1 \end{pmatrix}$ over a field $\F$.
\item $S : \R^2 \longrightarrow \R^2$, such that $S \begin{bmatrix} x \\ y\end{bmatrix} =  \begin{bmatrix} y \\ -x\end{bmatrix}$.
  \item $T : \C^2 \longrightarrow \C^2$, such that $T \begin{bmatrix} x \\ y\end{bmatrix} =  \begin{bmatrix} y \\ -x\end{bmatrix}$.
  \eabc

\item Let $D \in \mathscr{L}(C^\infty(\R))$ be the differentiation linear operator, that is
  $D(y(t)) = y'(t)$.  In this problem,
  you may use the fact that if $p(x) \in \PP(\R)$ has degree $d$, then
  $\dim \ker p(D) = d$.
  \babc
\item Show that $(D-rI)(f(t)e^{rt}) = f'(t) e^{rt}$.
\item Find a basis for $\ker (D-2I)^3$ (hint: try functions of the form $y(t) = f(t) e^{rt}$).
  \item Find a basis for $\ker ((D-I)(D-2I)^2)$.
  \item The solution set of the functions $y(t)$ which satisfy the differential equation $y''' + y'' - y' + y = 0$
    is a vector space.  Find a basis for this vector space.
  \eabc

  \pagebreak  
\item {\bf The Fibonacci numbers}.  Recall that the fibonacci numbers are $F_1 = 1$,
  $F_2 = 1$, and $F_n = F_{n-2} + F_{n-1}$, for $n \ge 3$.  So $F_3 = 2$ and $F_4 = 3$,
  and $F_{100} = 354224848179261915075$.
\babc
\item Find a $2 \times 2$ matrix $A$ such that
  \[
  A^n \begin{bmatrix}
    0 \\ 1
  \end{bmatrix}
  =
\begin{bmatrix}
    F_{n} \\ F_{n+1}
  \end{bmatrix},
\]
for all $n \ge 1$.
\item Find the eigenvalues of this matrix $A$.
\item Find a basis of $\R^2$ consisting of eigenvectors of this matrix $A$.
\item Use part (c) to compute   $A^n \begin{bmatrix}
    0 \\ 1
\end{bmatrix}$.  Use this to find a ``closed-form'' formula for $F_n$.
\item Use your description in part (d) to show that $F_n$ is the integer closest to
  \[ \frac{1}{\sqrt{5}} \left( \frac{1+\sqrt{5}}{2} \right)^n. \]
\eabc


  \item Consider the stochastic matrix
  \[ A = \begin{pmatrix}
    0.6 & 0.1 & 0.1 \\
    0.1 & 0.9 & 0.2 \\
    0.3 & 0   & 0.7
  \end{pmatrix}.
  \]
  \babc 
\item Is $A$ a regular stochastic matrix?
\item If $A$ is the transition matrix for a 3 state Markov chain, and the initial probability vector (at time step 0) is
  \[
  P = \begin{pmatrix} 0.3 \\ 0.3 \\ 0.4 \end{pmatrix},
  \]
  Find the proportions in each state after 2 time steps.  Find the eventual proportions in each state, (i.e. the limiting proportions,
  as the time step goes to infinity), by computing the fixed probability vector.
  \eabc


\item The vector space of polynomials $\PP(\F)$ has a multiplication on it too (we can multiply 2 polynomials, and get a new polynomial).
  This makes $\PP(\F)$ into what is called an algebra).
  An {\em ideal} is a subspace $I \subseteq \PP(\F)$ which satisfies the property that
  if $h(x) \in \PP(\F)$, and $f(x) \in I$, then $h(x)f(x) \in I$.
  \babc
\item Fix a polynomial $f(x)$.  Show that the set
  \[ \langle f(x) \rangle = \left\{\, h(x)f(x) \in \PP(\F) \mid \mbox{$h(x) \in \PP(\F)$}\right\}\ \]
  is an ideal in $\PP(\F)$ (called the {\em ideal generated by $f(x)$}).
\item Suppose that the ideal $I$ contains a polynomial $g(x)$ of degree $d$, but no
  nonzero polynomial of lower degree.  Show that every polynomial in $I$ is divisible by $g(x)$.
  Use this to show that $I = \langle g(x) \rangle$.
  \eabc

\item If $f(x)$ is a polynomial, and $A$ is an $n \times n$ matrix, we can form the $n \times n$ matrix
  $f(A)$.  For example, if $f(x) = x^2-2x+3$, then $f(A) = A^2 - 2A + 3I_n$.
\babc
\item  Show that if $A$ is an $n \times n$ matrix, then \[ ann(A) := \{ f(x) \in \PP(\F) \mid f(A) = 0_{n \times n} \} \]
  is an ideal in $\PP(\F)$ is called the annihilator of $A$.


\item Find $ann(A)$, if
  \[ A = \begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix}. \]
    \eabc
  
  \item
      Suppose that $A \in \F^{m \times n}$, and let $T = L_A : \F^n \longrightarrow \F^m$ be the corresponding
      linear map. We often represent elements of $\F^n$ as column vectors.
      Elements of ${\F^n}^* = \LL(\F^n, \F^1)$ can be thought of as $1 \times n$ matrices, i.e. row vectors.

      A matrix $A$ has 4 interesting subspaces
      associated with it:
      \babc
      \item[(i)] $colspan(A) \subset \F^m$ is the span of the columns of $A$,
      \item[(ii)] $rowspan(A) \subset {\F^n}^*$ is the span of the rows of $A$,
      \item[(iii)] $\ker(A) = \{ {\bf x} \mid           A{\bf x} = {\bf 0} \}$,
        \item[(iv)]
          the left kernel of $A$,
            $leftker(A) = \{ {\bf y} \in {\F^m}^* \mid {\bf y} A = {\bf 0}\}$.
            \eabc

      \noindent
      (Note, for both parts of this problem, let $T = L_A$).  
            \babc
          \item If ${\bf w} = (a_1, \ldots, a_n)$ is a row vector (i.e. a $n \times 1$ matrix)
            representing an element of ${\F^n}^* = \LL(\F^n, \F^1)$,
%              which represents the element
%              $\varphi = a_1 \varphi_1 + \ldots + a_n \varphi_n \in {\F^n}^*$,
              and
            if ${\bf v} = b_1 e_1 + \ldots + b_n e_n \in \F^n$ is a column vector,
            what is $\varphi({\bf v})$?
          \item On the last homework, several subspaces are mentioned:
            \[\ker(T^*), \ker(T), image(T), image(T^*), (\ker(T))^o, (image(T))^o. \]
            Some of these are the same as the 4 subspaces associated to $A$ above.
            Match them up, and prove that they are the same.
      \eabc

\end{enumerate}

\end{document}



